Scaling or Thinning Bones
An Armature never shows up during a render. If it sticking out is annoying while modeling, you can choose between five options for its shape. To change, select the armature and go to the Properties panel > Skeleton tab > Display dropdown and choose. I would describe the five, but it would take a while, so I'll just say that I believe Stick is the smallest.



Note that the image is a cropped version of one belonging to chichige bobo in this YouTube video.

Lastly, just thought I'd add that Octahedral bones (what is currently displaying) do stretch out like in your third image, but only when using a Stretch To Constraint. You would not want to employ this for just basic rigging.